Appell's equation of motion

In classical mechanics, 'Appell's equation of motion is an alternative general formulation of classical mechanics. Described by Paul Émile Appell in 1900, it is summarized in the equation

${\frac {\partial S}{\partial q_{r}}}=G_{r}$

where q_r is an arbitrary generalized coordinate and G_r is its corresponding generalized force; that is, the work done is given by

$dW=\sum _{r=1}^{s}G_{r}dq_{r}$

The function S is the mass-weighted sum of the particle accelerations squared

$S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\left|\mathbf {a} _{k}\right|^{2}$

where the index k runs over the N particles. Although fully equivalent to the other formulations of classical mechanics such as Newton's second law and the principle of least action, Appell's equation of motion may be more convenient in some cases, particularly when constraints are involved. Appell’s formulation can be viewed as a variation of Gauss' principle of least constraint.

Example: Euler's equations

Euler's equations provide an excellent illustration of Appell's formulation.

Consider a rigid body of N particles joined by rigid rods. The rotation of the body may be described by an angular velocity vector ${\boldsymbol {\omega }}$, and the corresponding angular acceleration vector

${\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}$

The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation $\delta {\boldsymbol {\phi }}$ is $dW=\mathbf {N} \cdot \delta {\boldsymbol {\phi }}$. The velocity of the k^th particle is given by

$\mathbf {v} _{k}={\boldsymbol {\omega }}\times \mathbf {r} _{k}$

where r_k is the particle's position in Cartesian coordinates; its corresponding acceleration is

$\mathbf {a} _{k}={\frac {d\mathbf {v} _{k}}{dt}}={\boldsymbol {\alpha }}\times \mathbf {r} _{k}+{\boldsymbol {\omega }}\times \mathbf {v} _{k}$

Therefore, the function S may be written as

$S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\left(\mathbf {a} _{k}\cdot \mathbf {a} _{k}\right)={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\left\{\left({\boldsymbol {\alpha }}\times \mathbf {r} _{k}\right)^{2}+\left({\boldsymbol {\omega }}\times \mathbf {v} _{k}\right)^{2}+2\left({\boldsymbol {\alpha }}\times \mathbf {r} _{k}\right)\cdot \left({\boldsymbol {\omega }}\times \mathbf {v} _{k}\right)\right\}$

Setting the derivative of S with respect to ${\boldsymbol {\alpha }}$ equal to the torque yields Euler's equations

$I_{xx}\alpha _{x}-\left(I_{yy}-I_{zz}\right)\omega _{y}\omega _{z}=N_{x}$

$I_{yy}\alpha _{y}-\left(I_{zz}-I_{xx}\right)\omega _{z}\omega _{x}=N_{y}$

$I_{zz}\alpha _{z}-\left(I_{xx}-I_{yy}\right)\omega _{x}\omega _{y}=N_{z}$

Derivation

The change in the particle positions r_k for an infinitesimal change in the s generalized coordinates is

$d\mathbf {r} _{k}=\sum _{r=1}dq_{r}{\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}$

Taking two derivatives with respect to time yields an equivalent equation for the accelerations

${\frac {\partial \mathbf {a} _{k}}{\partial \alpha _{r}}}={\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}$

The work done by an infinitesimal change dq_r in the generalized coordinates is

$dW=\sum _{r}G_{r}dq_{r}=\sum _{k=1}^{N}\mathbf {F} _{k}\cdot d\mathbf {r} _{k}=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot d\mathbf {r} _{k}$

Substituting the formula for dr_k and swapping the order of the two summations yields the formulae

$dW=\sum _{r}G_{r}dq_{r}=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \sum _{r=1}^{2}dq_{r}\left({\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}\right)=\sum _{r=1}^{2}dq_{r}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}\right)$

Therefore, the generalized forces are

$G_{r}=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}\right)=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {a} _{k}}{\partial \alpha _{r}}}\right)$

This equals the derivative of S with respect to the generalized accelerations

${\frac {\partial S}{\partial \alpha _{r}}}={\frac {\partial }{\partial \alpha _{r}}}{\frac {1}{2}}\sum _{k=1}^{N}m_{k}\left|\mathbf {a} _{k}\right|^{2}=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {a} _{k}}{\partial \alpha _{r}}}\right)$

yielding Appell’s equation of motion

${\frac {\partial S}{\partial \alpha _{r}}}=G_{r}$

Further reading

• Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed. ed.). New York: Dover Publications. ISBN. {{cite book}}: |edition= has extra text (help)

• Seeger (1930). "Unknown title". Journal of the Washington Academy of Science. 20: 481–?.

• Brell, H (1913). "Unknown title". Wien. Sitz. 122: 933–?. Connection of Appell's formulation with the principle of least action.